# Category of modules

In algebra, given a ring *R*, the **category of left modules** over *R* is the category whose objects are all left modules over *R* and whose morphisms are all module homomorphisms between left *R*-modules. For example, when *R* is the ring of integers **Z**, it is the same thing as the category of abelian groups. The **category of right modules** is defined in a similar way.

**Note:** Some authors use the term **module category** for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action.[1]

## Properties

The category of left modules (or that of right modules) is an abelian category. The category has enough projectives[2] and enough injectives.[3] Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules.

Projective limits and inductive limits exist in the category of (say left) modules.[4]

Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category.

## Example: the category of vector spaces

The category **K-Vect** (some authors use **Vect**_{K}) has all vector spaces over a fixed field *K* as objects and *K*-linear transformations as morphisms. Since vector spaces over *K* (as a field) are the same thing as modules over the ring *K*, **K-Vect** is a special case of **R-Mod**, the category of left *R*-modules.

Much of linear algebra concerns the description of **K-Vect**. For example, the dimension theorem for vector spaces says that the isomorphism classes in **K-Vect** correspond exactly to the cardinal numbers, and that **K-Vect** is equivalent to the subcategory of **K-Vect** which has as its objects the free vector spaces *K*^{n}, where *n* is any cardinal number.

## Generalizations

The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives).

## See also

- Algebraic K-theory (the important invariant of the category of modules.)
- Category of rings
- Derived category
- Module spectrum
- Category of graded vector spaces
- Category of abelian groups
- Category of representations

## References

- "module category in nLab".
*ncatlab.org*. - trivially since any module is a quotient of a free module.
- Dummit–Foote, Ch. 10, Theorem 38.
- Bourbaki, § 6.

- Bourbaki,
*Algèbre*; "Algèbre linéaire." - Dummit, David; Foote, Richard.
*Abstract Algebra*. - Mac Lane, Saunders (September 1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics.**5**(second ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001.